Least-Order Torsion-Gravity for Fermion Fields, and the Non-Linear Potentials in the Standard Models

We will consider the least-order torsional completion of gravity for a spacetime filled with fermionic Dirac matter fields, and we study the effects of the background-induced non-linear potentials for the matter field themselves in view of their effects for both standard models of physics: from the one of cosmology to that of particles, we will discuss the mechanisms of generation of the cosmological constant and particle masses as well as the phenomenology of leptonic weak-like forces and neutrino oscillations, the problem of zero-point energy, how there can be neutral massive fields as candidates for dark matter, and avoidance of gravitational singularity formation; we will show the way in which all these different effects can nevertheless be altogether described in terms of just a single model, which will be thoroughly discussed in the end.Comment: 11 page

Oscillation behavior of higher order functional differential equations with distributed deviating arguments

In this thesis we consider oscillatory and nonoscillatory behavior of functional differential equations and study third and n-th order functional differential equations qualitatively. Usually a qualitative approach is concerned with the behavior of solutions of a given differential equation and does not seek explicit solutions.;This dissertation is divided into five chapters. The first chapter consists of preliminary material which introduce well-known basic concepts. The second chapter deals with the oscillatory behavior of solutions of third order differential equations and functional differential equations with discrete and continuous delay of the form (bt(a t(x\u27 t)a)\u27 )\u27+qt fxt =rt, (bt(a t(x\u27 t)a)\u27 )\u27+qt fxgt =rt , (bt(( atx\u27 t)g)\u27 )\u27+(q1 txt) \u27+q2t x\u27t=h t, (bt(a tx\u27t )\u27)\u27+ i=1mqit f(x(sit ))=ht and (bt(a tx\u27t )\u27)\u27+ cdqt,x fxst,x dx=0. In chapter three we present sufficient conditions for oscillatory behavior of n-th order homogeneous neutral differential equation with continuous deviating arguments of the form at&sqbl0; xt+pt xtt &sqbr0;n-1 \u27+dcd qt,xf xst,x dx=0. Chapter four is devoted to n-th order neutral differential equation with forcing term of the form &sqbl0;xt+ i=1mpit x(tit )&sqbr0;n +l1a bq1t,x f1(x(s1 t,x))dx +l2ab q2t,xf 2(x(s2t,x ))dx=ht . Lastly, in chapter five we present sufficient conditions involving the coefficients and arguments only for n-th order neutral functional differential equation with constant coefficient of the form &sqbl0; xt+lax t+ah+mbxt+b g&sqbr0;n =pcdx t-xdx+qc dxt+x dx

Oscillation Criteria For Even Order Nonlinear Neutral Differential Equations With Mixed Arguments

This paper deals with the oscillation criteria for nth order nonlinear neutral  mixed type dierential equations

Oscillation of solutions to a higher-order neutral PDE with distributed deviating arguments

This article presents conditions for the oscillation of solutions to neutral partial differential equations. The order of these equations can be even or odd, and the deviating arguments can be distributed over an interval. We also extend our results to a nonlinear equation and to a system of equations

Statistical mechanics approach to some problems in conformal geometry

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in \bbR^n, or \bbS^n, n\in \bbN, which are distributed according to the configurational microcanonical measure $\delta(E-H)$, or rather some regularization thereof, where H is the configurational Hamiltonian and E the configurational energy. When $N\to\infty$ with non-extensive energy scaling E=N^2 \vareps, the particle positions become i.i.d. according to a self-consistent Boltzmann distribution, respectively a superposition of such distributions. The self-consistency condition in n dimensions is some nonlinear elliptic PDE of order n (pseudo-PDE if n is odd) with an exponential nonlinearity. When n=2, this PDE is known in statistical mechanics as Poisson-Boltzmann equation, with applications to point vortices, 2D Coulomb and magnetized plasmas and gravitational systems. It is then also known in conformal differential geometry, where it is the central equation in Nirenberg's problem of prescribed Gaussian curvature. For constant Gauss curvature it becomes Liouville's equation, which also appears in two-dimensional so-called quantum Liouville gravity. The PDE for n=4 is Paneitz' equation, and while it is not known in statistical mechanics, it originated from a study of the conformal invariance of Maxwell's electromagnetism and has made its appearance in some recent model of four-dimensional quantum gravity. In differential geometry, the Paneitz equation and its higher order n generalizations have applications in the conformal geometry of n-manifolds, but no physical applications yet for general n. Interestingly, though, all the Paneitz equations have an interpretation in terms of statistical mechanics.Comment: 17 pages. To appear in Physica

Oscillatory and nonoscillatory properties of solutions of functional differential equations and difference equations

Oscillation and nonoscillation of solutions of functional differential equations and difference equations are analyzed qualitatively. A qualitative approach is usually concerned with the behavior of solutions of a given equation and does not seek explicit solutions. The dissertation is divided into five chapters. The first chapter is essentially introductory in nature. Its main purpose is to introduce certain well-known basic concepts and to present some result that are not as well-known. In chapter 2 and chapter 3 we present sufficient conditions for oscillation of solutions of neutral differential equations of the form [a(t)[x(t) + p(t)x([tau](t))] [superscript](n-1)] [superscript]\u27 + q(t)f(x([sigma](t))) = 0and [x(t) + p(t)x([tau](t))] [superscript](n) + q[subscript]1(t)f(x([sigma][subscript]1(t))) + q[subscript]2(t)f(x([sigma][subscript]2(t))) = h(t)respectively. Chapter 4 discusses the oscillation, nonoscillation, and the asymptotic behavior of solutions of higher order functional differential equations of the form (r[subscript]2(r[subscript]1 x[superscript]\u27(t))[superscript]\u27)[superscript]\u27 + q(t)f(x([sigma](t))) = h(t)and x[superscript](n)(t) + F(t,x([sigma][subscript]1(t)),...,x([sigma][subscript]m(t))) = h(t).Chapter 5 is devoted the study of oscillatory solutions of neutral type difference equations of the form [delta][a[subscript]n[delta][superscript]m-1(x[subscript]n + p[subscript]nx[subscript][tau][subscript]n)] + q[subscript]nf(x[subscript][sigma][subscript]n) = 0and that of asymptotic behavior for n → [infinity] of solutions of equations of the form [delta][superscript]mx[subscript]n + F(n, x[subscript][sigma][subscript]n) = h[subscript]n.The results obtained here are the discrete analogs of several of those in chapter 1 and chapter 4;A function x(t) : [a,[infinity]) → R is said to be oscillatory if it has a zero on [T,[infinity]) for every T ≥ a; otherwise it is called nonoscillatory. Similarly a sequence \x[subscript]n of real numbers is oscillatory if it is not eventually positive or eventually negative; otherwise it is nonoscillatory

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes.Comment: 28 pages, 3 figures. Main result strengthened by removing a smallness condition. Limiting case of the black soliton now postponed to a companion pape